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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | ⊢ 𝐾 = (mTC‘𝑇) |
mexval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mexval.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mexval | ⊢ 𝐸 = (𝐾 × 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 ⊢ 𝐸 = (mEx‘𝑇) | |
2 | fveq2 6645 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
4 | 2, 3 | eqtr4di 2851 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) |
5 | fveq2 6645 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6 | eqtr4di 2851 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) |
8 | 4, 7 | xpeq12d 5550 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) |
9 | df-mex 32847 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
10 | fvex 6658 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
11 | fvex 6658 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
12 | 10, 11 | xpex 7456 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V |
13 | 8, 9, 12 | fvmpt3i 6750 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
14 | xp0 5982 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
15 | 14 | eqcomi 2807 | . . . 4 ⊢ ∅ = (𝐾 × ∅) |
16 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
17 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
18 | 6, 17 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
19 | 18 | xpeq2d 5549 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) |
20 | 15, 16, 19 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
21 | 13, 20 | pm2.61i 185 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) |
22 | 1, 21 | eqtri 2821 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 × cxp 5517 ‘cfv 6324 mTCcmtc 32824 mRExcmrex 32826 mExcmex 32827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-mex 32847 |
This theorem is referenced by: mexval2 32863 msubff 32890 msubco 32891 msubff1 32916 mvhf 32918 msubvrs 32920 |
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